Moduloid - Abelian Unital Magma
1. Introduction
In the modular arithmetic, the set of remainders in division forms a quotient group. For example, the group Z/3Z is composed of the elements: 0, 1, and 2. It is expressed graphically as,
Namely in this case, the ends of segment shown above are regarded as the same point. Naturally such visualization of quotient group is extended to that of multidimensional quotient group. For example, the group Z/3Z x Z/2Z is expressed as,
where the points having the same label are regarded as the same point. The above graphics suggests the similarity with the feature of torus. Actually two dimensional torus group is represented as R/Z x R/Z and graphically expressed as,
Of course Z/3Z x Z/2Z and R/Z x R/Z are completely different groups. However the former is thought to characterize the feature of two dimensional torus in a certain meaning. For example, when the two dimensional torus is "approximated" practically to be dealt with by computer as Z/300Z x Z/200Z, that group has the properties similar to Z/3Z x Z/2Z since they are homomorphic. In the following discussion, we try to characterize topological space by finite quotient set induced from generalized modulo operation.
2. Real projective plane, sphere, Klein bottle, et al.
The real projective plane as quotient space is represented as,
Discretizing it, we obtain the following expression.
We denote it here as (Mp[2,2], +p) where Mp[2,2] is the set of all distinctive elements appeared on the above expression and +p is the addition operator defined on Mp[2,2]. From the above expression, the Cayley table is composed as,
| +p |
(0 0) |
(1 0) |
(0 1) |
(1 1) |
(2 0) |
| (0 0) |
(0 0) |
(1 0) |
(0 1) |
(1 1) |
(2 0) |
| (1 0) |
(1 0) |
(2 0) |
(1 1) |
(0 1) |
(1 0) |
| (0 1) |
(0 1) |
(1 1) |
(2 0) |
(1 0) |
(0 1) |
| (1 1) |
(1 1) |
(0 1) |
(1 0) |
(0 0) |
(1 1) |
| (2 0) |
(2 0) |
(1 0) |
(0 1) |
(1 1) |
(0 0) |
It should be noted that Mp does not form the group under the operation +p since (0 1) and (1 0) don't have their inverse elements. Here we call generally the form (M, +) a "moduloid". Since (Mp[2,2], +p) satisfies the associative law and has the unit element (0, 0), it is a monoid. Generally the set of elements of monoid after eliminating the elements which don't have corresponding inverse elements forms the maximum subgroup in that monoid. In this case the Cayley table for such group is expressed as,
| +p |
(0 0) |
(1 1) |
(2 0) |
| (0 0) |
(0 0) |
(1 1) |
(2 0) |
| (1 1) |
(1 1) |
(0 0) |
(1 1) |
| (2 0) |
(2 0) |
(1 1) |
(0 0) |
It also should be noted when the moduloid for the above quotient space before discretizing is expressed as (Mp(a, b), +p), Mp(a, b) does not form group with the operation +p. For, if Mp(a, b) is supposed to be a group, then an element (x, y) which is the inverse of (c, 0) (0 < c < b) must satisfy the folwlowing equation;
(c, 0) +p (x, y) = (b, a) (» (0, 0))
namely,
(x, y) = (b-c, a) » (c, 0)
However,
(c, 0) +p (c, 0) = (2c, 0) ¹ (0, 0)
This contradiction indicates Mp(a, b) is not group.
The sphere is represented with the quotient space as,
Discretizing it as,
we obtain the following Cayley table.
| +s |
(0 0) |
(0 1) |
(1 0) |
(1 1) |
(0 2) |
(2 0) |
| (0 0) |
(0 0) |
(0 1) |
(1 0) |
(1 1) |
(0 2) |
(2 0) |
| (0 1) |
(0 1) |
(0 2) |
(1 1) |
(0 1) |
(0 1) |
(1 0) |
| (1 0) |
(1 0) |
(1 1) |
(2 0) |
(1 0) |
(0 1) |
(1 0) |
| (1 1) |
(1 1) |
(0 1) |
(1 0) |
(0 0) |
(1 1) |
(1 1) |
| (0 2) |
(0 2) |
(0 1) |
(0 1) |
(1 1) |
(0 0) |
(0 0) |
| (2 0) |
(2 0) |
(1 0) |
(1 0) |
(1 1) |
(0 0) |
(0 0) |
Here we denote that moduloid as (Ms[2,2], +s). It should be noted that Ms does not form group under the operation +s but forms a monoid. The maximum subgroup in that monoid is expressed as,
| +s |
(0 0) |
(1 1) |
(0 2) |
(2 0) |
| (0 0) |
(0 0) |
(1 1) |
(0 2) |
(2 0) |
| (1 1) |
(1 1) |
(0 0) |
(1 1) |
(1 1) |
| (0 2) |
(0 2) |
(1 1) |
(0 0) |
(0 0) |
| (2 0) |
(1 0) |
(1 1) |
(0 0) |
(0 0) |
The Klein bottle as quotient space is represented as,
Discretizing it as,
we obtain the following Cayley table.
| +k |
(0 0) |
(0 1) |
(1 0) |
(1 1) |
(2 0) |
(2 1) |
| (0 0) |
(0 0) |
(0 1) |
(1 0) |
(1 1) |
(2 0) |
(2 1) |
| (0 1) |
(0 1) |
(0 0) |
(1 1) |
(2 0) |
(2 1) |
(1 0) |
| (1 0) |
(1 0) |
(1 1) |
(2 0) |
(2 1) |
(0 0) |
(0 1) |
| (1 1) |
(1 1) |
(2 0) |
(2 1) |
(1 0) |
(0 1) |
(0 0) |
| (2 0) |
(2 0) |
(2 1) |
(0 0) |
(0 1) |
(1 0) |
(1 1) |
| (2 1) |
(2 1) |
(1 0) |
(0 1) |
(0 0) |
(1 1) |
(2 0) |
We denote it as (Mk[3,2],+k). In this time Mk[3,2] forms a group under the operation +k.
As the application of torus and Klein bottle, the following quotient space;
is discretized as,
The following applet shows the Cayley table of the moduloids for torus, sphere, real projective plane, and Klein bottle. After selecting the type of moduloid (i.e. "Torus", "Sphere", "Projective", or "Klein Btl") and the number of divisions for segment, click "Start". Then the corresponding Cayley table is shown. On the table the unit elements (0, 0) are specially indicated with red.
3. Chaotic space
So far we've dealt with the quotient spaces which consist of the pairs of equivalent segments on the rectangle. The moduloids appeared in those cases were either groups or monoids. Now we consider the following quotient space.
In this case there are three segments which are equivalent each other on the rectangle. Practically the addition operation +c on that quotient space before discretizing is defined as,
(a,c)+c(b,d)
= (a+b, c+d) ...(0 £ a+b < 3 and 0 £ c+d < 3) or
= (a+b-3, c+d) ...(3 £ a+b and 0 £ c+d < 3) or
= (2(a+b), c+d-3) ...(0 £ a+b < 1.5 and 3 £ c+d) or
= (3-2(a+b-1.5), c+d-3) ...(1.5 £ a+b < 3 and 3 £ c+d) or
= (2(a+b-3), c+d-3) ...(3 £ a+b < 4.5 and 3 £ c+d) or
= (3-2(a+b-4.5), c+d-3) ...(4.5 £ a+b and 3 £ c+d)
On this space there are the pair of solutions of ordinary differential equations which fuses after time elapsing. For example, the solution of the following equation;
dx/dt = 0 (x(0) = 1.2)
dy/dt = 1.5 (y(0) = 0)
coincides with the solution starting from x=1.8 after t=2 and shows chaotic behaviour (® remark). In the following figures, the upper-left is the trajectory of the solution starting from x=1.2 before reaching t=2 and the upper-right is that of the solution starting from x=1.8. The lower figure is that of the fused solution after t=2.
Here we call the quotient space in which the solution of an ordinary differential equation; dx/dt = c (c: constant vector) shows chaotic behaviour a "chaotic space". Discretizing the space shown above as,
we obtain the following Cayley table.
| +c |
(0 0) |
(0 1) |
(0 2) |
(1 0) |
(2 0) |
(1 1) |
(1 2) |
(2 1) |
(2 2) |
| (0 0) |
(0 0) |
(0 1) |
(0 2) |
(1 0) |
(2 0) |
(1 1) |
(1 2) |
(2 1) |
(2 2) |
| (0 1) |
(0 1) |
(0 2) |
(0 0) |
(1 1) |
(2 1) |
(1 2) |
(2 0) |
(2 2) |
(2 0) |
| (0 2) |
(0 2) |
(0 0) |
(0 1) |
(1 2) |
(2 2) |
(2 0) |
(2 1) |
(2 0) |
(2 1) |
| (1 0) |
(1 0) |
(1 1) |
(1 2) |
(2 0) |
(0 0) |
(2 1) |
(2 2) |
(0 1) |
(0 2) |
| (2 0) |
(2 0) |
(2 1) |
(2 2) |
(0 0) |
(1 0) |
(0 1) |
(0 2) |
(1 1) |
(1 2) |
| (1 1) |
(1 1) |
(1 2) |
(2 0) |
(2 1) |
(0 1) |
(2 2) |
(2 0) |
(0 2) |
(0 0) |
| (1 2) |
(1 2) |
(2 0) |
(2 1) |
(2 2) |
(0 2) |
(2 0) |
(2 1) |
(0 0) |
(0 1) |
| (2 1) |
(2 1) |
(2 2) |
(2 0) |
(0 1) |
(1 1) |
(0 2) |
(0 0) |
(1 2) |
(2 0) |
| (2 2) |
(2 2) |
(2 0) |
(2 1) |
(0 2) |
(1 2) |
(0 0) |
(0 1) |
(2 0) |
(2 1) |
We denote that moduloid as (Mc[3,3],+c). Since the operation +c does not satisfy the associative law (ex. ((0,2)+c(0,1))+c(1,2)=(0,0)+c(1,2)=(1,2) vv (0,2)+c((0,1)+c(1,2))=(0,2)+c(2,0)=(2,2)), this moduloid is not monoid. Generally when the number of equivalent segments in the rectangle is greater than two, the moduloid is always not monoid.
4. In higher dimension
The method described above is easily extended to three dimensional or higher dimensional space. For examples, a three dimensional torus is represented with quotient space as,
also the three dimensional sphere is represented as,
where any two areas surrounded with the equivalent segments with the same order and directions are regarded as the equivalent areas. And these quotient spaces are discretized to obtain the corresponding moduloids with the manner similar to that described in the former sections. One of the significant points is there is the three dimensional chaotic space in which the solution of ordinary differential equation does not coincides with any other solutions as long as starting from different initial position from others contrary to the case of two dimensional chaotic space. For example, in the following quotient space;
there is a continuous map from the bottom square area to the top square area in the above figure which has the differential equation dx/dt = (0, 0, 1) a chaotic behaviour. To see it, firstly we think the set of continuous maps S from bottom to top square area as shown in the following figure.
Let (c) an area in the region A. As shown in the lower-second figure, there is the set of continuous maps S' in S which maps a part of (c) to the area (a) in the region B. Also there is the set of continuous maps S'' in S' which maps (a) to (b) in the region A as shown in the lower-third figure. Finally there is the set of continuous maps S''' in S'' which maps (b) to an area in (c) as shown in the lower-right figure. Namely when denoting a map in S''' as f, f3 forms a horseshoe map. Therefore the solutions of dx/dt = (0, 0, 1) show chaotic behaviour in that quotient space. More practically let S* the set of continuous maps which maps the bottom square area to the top one as shown in the following figure;
then there is a map f in S* such that f6 forms a horseshoe map as shown in the above figure. In this case the associative law of addition is satisfied and that moduloid forms a monoid. In general we can not know whether the given three dimensional quotient space is the chaotic space or not from the form of its moduloid contrary to the case of two dimensional chaotic space.
However it should be noted that there is the three dimensional chaotic space which is not characterized by the horseshoe map. To see it, firstly we consider the following differential equations.
dx/dt = {-0.2-4(z-1)}x+y+(x2+y2)·0.103x
dy/dt = -x
dz/dt = 0.1·(x2+y2-z)
The trajectory of the above equations shows chaotic behaviour under certain initial conditions as shown in the following figure (stereograph / view: parallel eyes).
(simulation)
Now we define the mapping f from a z-plane to itself based on that differential equation system as,
f: {(x, y, z); z = c} ® {(x, y, z); z = c} (c: constant)
where f(x, y, z) is the solution of the above differential equation system starting from (x, y, z) and turning back to the original plane {(x, y, z); z = c}. As shown in the following figure, some solutions may diverge and never turn back to that plane.
Let h = f2. Namely h(x, y,z) is the solution of the above differential equation system starting from (x, y, z) and turning back to the original plane {(x, y, z); z = c} at the second time as shown in the following figure.
Then h maps a certain ring on the z-plane: z = 1 to a deformed ring on the same z-plane as,
The following figure shows how the depicted yellow region is transformed by the mapping h.
As you imagine from the above figure, particularly the crevice-like part in the image would fold down complicatedly through the iteration of h. The following figures are the results of simulation of the mapping on a circle in the ring on the z-plane.
In the figures, the cyan curve is the image of red circle by h (left), h2 (middle), and h3 (right). Although this mapping is different from the horseshoe mapping, it is shown the automorphic points on the z-plane by h (i.e. the points h(x, y, z) where (x, y, z) resides in the plane z =c) form a fractal set. The following figures are the sets of automorphic points by h on z = 0.6 (upper-left), z = 0.8 (upper-right), z = 1 (middle-left), z = 1.2 (middle-right), and z = 1.4 (lower).
When the space composed of automorphic points characterised by the mapping h is represented with the quotient space as,
then the differential equation system showing chaotic behaviour on this quotient space is written simply as,
dx/dt = (0, 0, 1)
Therefore that quotient space is the chaotic space.
(For details of the dynamical system mentioned above, visit here.)
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